View source for Dual graviton

{{Short description|Hypothetical particle found in supergravity}}
{{Infobox Particle
| bgcolour =
| name = Dual graviton
| image =
| caption =
| num_types =
| composition = [[Elementary particle]]
| statistics = 
| group = [[Gauge boson]]
| generation =
| interaction = [[Gravitation]]
| particle =
| antiparticle = Self
| status = Hypothetical
| theorized = 2000s<ref name="Hull">
{{cite journal
 |last=Hull |first=C. M.
 |date=2001
 |title=Duality in Gravity and Higher Spin Gauge Fields
 |journal=[[Journal of High Energy Physics]]
 |volume=2001 |issue=9 |page=27
 |arxiv=hep-th/0107149
 |bibcode=2001JHEP...09..027H
 |doi=10.1088/1126-6708/2001/09/027
 |doi-access=free
}}</ref><ref name="Bekaert">
{{cite journal
 |last1=Bekaert |first1=X.
 |last2=Boulanger |first2=N.
 |last3=Henneaux |first3=M.
 |date=2003
 |title=Consistent deformations of dual formulations of linearized gravity: A no-go result
 |journal=[[Physical Review D]]
 |volume=67 |issue=4 |page=044010
 |arxiv=hep-th/0210278
 |bibcode=2003PhRvD..67d4010B
 |doi=10.1103/PhysRevD.67.044010
|s2cid=14739195
 }}</ref>
| discovered = 
| symbol = 
| mass = 
| mean_lifetime = 
| decay_particle =
| electric_charge = 0&nbsp;[[elementary charge|''e'']]
| color_charge =
| spin = 2
}}
{{string theory}}
{{Beyond the Standard Model}}
In [[theoretical physics]], the '''dual graviton''' is a hypothetical [[elementary particle]] that is a dual of the [[graviton]] under [[electric-magnetic duality]], as an [[S-duality]], predicted by some formulations of [[eleven-dimensional supergravity]].<ref name="deWit">
{{cite journal
 |last1=de Wit |first1=B.
 |last2=Nicolai |first2=H.
 |date=2013
 |title=Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions
 |journal=[[Journal of High Energy Physics]]
 |volume=2013 |issue=5 |page=77
 |arxiv=1302.6219
 |bibcode=2013JHEP...05..077D
 |doi=10.1007/JHEP05(2013)077
|s2cid=119201330
 }}</ref>

The dual graviton was first [[Curtright field|hypothesized]] in 1980.<ref name="Curtright">
{{cite journal
 |last=Curtright |first=T.
 |date=1985
 |title=Generalised Gauge Fields
 |journal=[[Physics Letters B]]
 |volume=165 |issue=4–6 |page=304
 |bibcode=1985PhLB..165..304C
 |doi=10.1016/0370-2693(85)91235-3
}}</ref> It was theoretically modeled in 2000s,<ref name="Hull" /><ref name="Bekaert" /> which was then predicted in eleven-dimensional mathematics of SO(8) [[supergravity]] in the framework of electric-magnetic duality.<ref name="deWit"/> It again emerged in the ''E''<sub>11</sub> generalized geometry in eleven dimensions,<ref name="West">
{{cite journal
 |last=West |first=P.
 |date=2012
 |title=Generalised geometry, eleven dimensions and ''E''<sub>11</sub>
 |journal=[[Journal of High Energy Physics]]
 |volume=2012 |issue=2 |pages=18
 |arxiv=1111.1642
 |bibcode=2012JHEP...02..018W
 |doi=10.1007/JHEP02(2012)018
|s2cid=119240022
 }}</ref> and the ''E''<sub>7</sub> generalized vielbein-geometry in eleven dimensions.<ref name="Godazgar">
{{cite journal
 |last1=Godazgar |first1=H.
 |last2=Godazgar |first2=M.
 |last3=Nicolai |first3=H.
 |date=2014
 |title=Generalised geometry from the ground up
 |journal=[[Journal of High Energy Physics]]
 |volume=2014 |issue=2 |pages=75
 |arxiv=1307.8295
 |bibcode=2014JHEP...02..075G
 |doi=10.1007/JHEP02(2014)075
 |doi-access=free
}}</ref> While there is no local coupling between graviton and dual graviton, the field introduced by dual graviton may be coupled to a [[BF model]] as non-local gravitational fields in extra dimensions.<ref name="Bizdadea">
{{cite journal
 |last1=Bizdadea |first1=C.
 |last2=Cioroianu |first2=E. M.
 |last3=Danehkar |first3=A.
 |last4=Iordache |first4=M.
 |last5=Saliu |first5=S. O.
 |last6=Sararu |first6=S. C.
 |date=2009
 |title=Consistent interactions of dual linearized gravity in ''D''&nbsp;=&nbsp;5: couplings with a topological BF model
 |journal=[[European Physical Journal C]]
 |volume=63 |issue=3 |pages=491–519
 |arxiv=0908.2169
 |bibcode=2009EPJC...63..491B
 |doi=10.1140/epjc/s10052-009-1105-0
|s2cid=15873396
 }}</ref>

A ''massive'' dual gravity of Ogievetsky–Polubarinov model<ref>{{Cite journal|last1=Ogievetsky|first1=V. I|last2=Polubarinov|first2=I. V|date=1965-11-01|title=Interacting field of spin 2 and the einstein equations|journal=Annals of Physics|volume=35|issue=2|pages=167–208|doi=10.1016/0003-4916(65)90077-1|issn=0003-4916|bibcode=1965AnPhy..35..167O}}</ref> can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.<ref name="Alshal 63">{{Cite journal|last1=Alshal|first1=H.|last2=Curtright|first2=T. L.|date=2019-09-10|title=Massive dual gravity in N spacetime dimensions|journal=Journal of High Energy Physics|language=en|volume=2019|issue=9|pages=63|doi=10.1007/JHEP09(2019)063|issn=1029-8479|bibcode=2019JHEP...09..063A|arxiv=1907.11537|s2cid=198953238}}</ref><ref name="Curtright 114777">{{Cite journal|last1=Curtright|first1=T. L.|last2=Alshal|first2=H.|date=2019-10-01|title=Massive dual spin 2 revisited|journal=Nuclear Physics B|volume=948|pages=114777|doi=10.1016/j.nuclphysb.2019.114777|issn=0550-3213|bibcode=2019NuPhB.94814777C|arxiv=1907.11532|s2cid=198953158}}</ref>

The previously mentioned theories of dual graviton are in flat space. In [[De Sitter space|de Sitter]] and [[Anti-de Sitter space|anti-de Sitter]] spaces (A)dS, the massless dual graviton exhibits less gauge symmetries dynamics compared with those of [[Curtright field]] in flat space, hence the mixed-symmetry field propagates in more degrees of freedom.<ref name=":0">{{Cite journal|last1=Boulanger|first1=N.|last2=Campoleoni|first2=A.|last3=Cortese|first3=I.|date=July 2018|title=Dual actions for massless, partially-massless and massive gravitons in (A)dS|journal=Physics Letters B|language=en|volume=782|pages=285–290|doi=10.1016/j.physletb.2018.05.046|bibcode=2018PhLB..782..285B|arxiv=1804.05588|s2cid=54826796}}</ref> However, the dual graviton in (A)dS transforms under GL(D) representation, which is identical to that of massive dual graviton in flat space.<ref>{{Cite journal|last1=Basile|first1=Thomas|last2=Bekaert|first2=Xavier|last3=Boulanger|first3=Nicolas|date=2016-06-21|title=Note about a pure spin-connection formulation of general relativity and spin-2 duality in (A)dS|journal=Physical Review D|language=en|volume=93|issue=12|pages=124047|doi=10.1103/PhysRevD.93.124047|issn=2470-0010|bibcode=2016PhRvD..93l4047B|arxiv=1512.09060|s2cid=55583084}}</ref> This apparent paradox can be resolved using the unfolding technique in Brink, Metsaev, and Vasiliev conjecture.<ref>{{Cite journal|last1=Brink|first1=L.|last2=Metsaev|first2=R.R.|last3=Vasiliev|first3=M.A.|date=October 2000|title=How massless are massless fields in AdS|journal=Nuclear Physics B|language=en|volume=586|issue=1–2|pages=183–205|doi=10.1016/S0550-3213(00)00402-8|bibcode=2000NuPhB.586..183B|arxiv=hep-th/0005136|s2cid=119512854}}</ref><ref>{{Cite journal|last1=Basile|first1=Thomas|last2=Bekaert|first2=Xavier|last3=Boulanger|first3=Nicolas|date=May 2017|title=Mixed-symmetry fields in de Sitter space: a group theoretical glance|journal=Journal of High Energy Physics|language=en|volume=2017|issue=5|pages=81|doi=10.1007/JHEP05(2017)081|issn=1029-8479|bibcode=2017JHEP...05..081B|arxiv=1612.08166|s2cid=119185373}}</ref> For the massive dual graviton in (A)dS, the flat limit is clarified after expressing dual field in terms of the [[Stueckelberg action|Stueckelberg]] coupling of a massless spin-2 field with a [[Proca action|Proca]] field.<ref name=":0" />

==Dual linearized gravity==

The dual formulations of linearized gravity are described by a mixed Young symmetry tensor <math>T_{\lambda_1 \lambda_2 \cdots \lambda_{D-3} \mu}</math>, the so-called dual graviton, in any spacetime dimension ''D'' > 4 with the following characters:<ref name="Bekaert" /><ref name="Danehkar">
{{cite journal
 |last=Danehkar |first=A.
 |date=2019
 |title=Electric-magnetic duality in gravity and higher-spin fields
 |journal=[[Frontiers in Physics]]
 |volume=6 |pages=146
 |bibcode=2019FrP.....6..146D
 |doi=10.3389/fphy.2018.00146
|doi-access=free
 }}</ref>
:<math>T_{\lambda_1 \lambda_2 \cdots \lambda_{D-3} \mu} = T_{[\lambda_1 \lambda_2 \cdots \lambda_{D-3}] \mu},</math>
:<math>T_{[\lambda_1 \lambda_2 \cdots \lambda_{D-3} \mu]} = 0.</math>
where square brackets show antisymmetrization.

For 5-D spacetime, the spin-2 dual graviton is described by the [[Curtright field]] <math>T_{\alpha\beta\gamma}</math>. The symmetry properties imply that
:<math>T_{\alpha\beta\gamma} = T_{[\alpha\beta]\gamma},</math>
:<math>T_{[\alpha\beta]\gamma}+T_{[\beta\gamma]\alpha}+T_{[\gamma\alpha ]\beta} = 0.</math>

The Lagrangian action for the spin-2 dual graviton <math>T_{\lambda_1 \lambda_2\mu}</math> in 5-D spacetime, the [[Curtright field]], becomes<ref name="Bekaert" /><ref name="Danehkar" />
:<math>{\cal L}_{\rm dual}=-\frac{1}{12}\left(F_{[\alpha\beta\gamma]\delta}F^{[\alpha\beta\gamma]\delta}-3F_{[\alpha\beta\xi]}{}^{\xi}F^{[\alpha\beta\lambda]}{}_{\lambda}\right),</math>
where <math>F_{\alpha\beta\gamma\delta}</math> is defined as
:<math>F_{[\alpha\beta\gamma]\delta}=\partial_{\alpha} T_{[\beta\gamma ]\delta}+\partial_{\beta}T_{[\gamma\alpha]\delta} +\partial_{\gamma}T_{[\alpha\beta]\delta},</math> 
and the gauge symmetry of the [[Curtright field]] is
:<math>\delta_{\sigma,\alpha} T_{[\alpha\beta]\gamma}=2(\partial_{[\alpha} \sigma_{\beta]\gamma} +\partial_{[\alpha}\alpha_{\beta]\gamma}-\partial_{\gamma}\alpha_{\alpha\beta}).</math> 
The dual [[Riemann curvature tensor]] of the dual graviton is defined as follows:<ref name="Bekaert" />
:<math>E_{[\alpha\beta\delta][\varepsilon\gamma]}\equiv\frac{1}{2}
 (\partial_{\varepsilon}F_{[\alpha\beta\delta]\gamma}-\partial_{\gamma}F_{[\alpha\beta\delta]\varepsilon}),</math>
and the dual [[Ricci curvature]] tensor and [[scalar curvature]] of the dual graviton become, respectively 
:<math>E_{[\alpha\beta]\gamma}= g^{\varepsilon\delta} E_{[\alpha\beta\delta][\varepsilon\gamma]},</math> 
:<math>E_{\alpha}=g^{\beta\gamma}E_{[\alpha\beta]\gamma}.</math> 
They fulfill the following Bianchi identities
:<math>\partial_{\alpha}(E^{[\alpha\beta]\gamma}+g^{\gamma[\alpha}E^{\beta]})= 0,</math>

where <math>g^{\alpha\beta}</math> is the 5-D spacetime metric.

== Massive dual gravity ==
In 4-D, the Lagrangian of the ''spinless'' ''massive'' version of the dual gravity is

<math>\mathcal{L^{\rm spinless}_{\rm dual, massive}} = -\frac{1}{2}u+\frac{1}{2}(v-gu)^2+\frac{1}{3}g(v-gu)^3 \sideset{_3}{_2}F(1,\frac{1}{2},\frac{3}{2};2,\frac{5}{2};-4g^2(v-gu)^2),</math>

where <math>V^{\mu}=\frac{1}{6}\epsilon^{\mu\alpha\beta\gamma}V_{\alpha\beta\gamma}~, v=V_{\mu}V^{\mu} \text{and} ~u=\partial_{\mu}V^{\mu}.</math><ref>{{Cite journal|last=Curtright|first=Thomas L.|date=2019-10-01|title=Massive dual spinless fields revisited|journal=Nuclear Physics B|volume=948|pages=114784|doi=10.1016/j.nuclphysb.2019.114784|issn=0550-3213|bibcode=2019NuPhB.94814784C|arxiv=1907.11530|s2cid=198953144}}</ref> The coupling constant <math>g/m</math> appears in the equation of motion to couple the trace of the conformally improved energy momentum tensor <math>\theta</math> to the field as in the following equation

<math>\left(\Box+m^2\right)V_{\mu}=\frac{g}{m}\partial_{\mu}\theta.</math>

And for the spin-2 massive dual gravity in 4-D,<ref name="Curtright 114777"/> the Lagrangian is formulated in terms of the [[Hessian matrix]] that also constitutes [[Horndeski's theory|Horndeski]] theory (Galileons/[[massive gravity]]) through

<math>\text{det} (\delta^\mu_\nu+\frac{g}{m}K^\mu_\nu)=1-\frac{1}{2}(g/m)^2K_\alpha^\beta K_\beta^\alpha+\frac{1}{3}(g/m)^3K_\alpha^\beta K_\beta^\gamma K_\gamma^\alpha+\frac{1}{8}(g/m)^4\left[(K_\alpha^\beta K_\beta^\alpha)^2-2K_\alpha^\beta K_\beta^\gamma K_\gamma^\delta K_\delta^\alpha\right],</math>

where <math>K_\mu^\nu=3 \partial_\alpha T_{[\beta\gamma]\mu}\epsilon^{\alpha\beta\gamma\nu}</math>.

So the zeroth interaction part, i.e., the third term in the Lagrangian, can be read as <math>K_\alpha^\beta \theta_\beta^\alpha</math> so the equation of motion becomes

<math>\left(\Box+m^2\right)T_{[\alpha\beta]\gamma}=\frac{g}{m}P_{\alpha\beta\gamma,\lambda\mu\nu}\partial^{\lambda}\theta^{\mu\nu},</math>

where the <math>P_{\alpha\beta\gamma,\lambda\mu\nu}=2\epsilon_{\alpha\beta\lambda\mu}\eta_{\gamma\nu}+\epsilon_{\alpha\gamma\lambda\mu}\eta_{\beta\nu}-\epsilon_{\beta\gamma\lambda\mu}\eta_{\alpha\nu}</math> is [[Young symmetrizer]] of such SO(2) theory.

For solutions of the massive theory in arbitrary N-D, i.e., Curtright field <math>T_{[\lambda_1\lambda_2...\lambda_{N-3}]\mu}</math>, the symmetrizer becomes that of SO(N-2).<ref name="Alshal 63"/>

==Dual graviton coupling with BF theory==

Dual gravitons have interaction with topological [[BF model]] in ''D'' = 5 through the following Lagrangian action<ref name="Bizdadea"/>
:<math>S_{\rm L} = \int d^{5}x( {\cal L}_{\rm dual} + {\cal L}_{\rm BF}).</math>
where
:<math>{\cal L}_{\rm BF}=Tr[\mathbf{B}\wedge \mathbf{F}]</math>
Here, <math>\mathbf{F}\equiv d\mathbf{A} \sim R_{ab}</math> is the [[curvature form]], and <math>\mathbf{B} \equiv e^{a} \wedge e^{b} </math> is the background field.

In principle, it should similarly be coupled to a BF model of gravity as the linearized Einstein–Hilbert action in ''D'' > 4:
:<math>S_{\rm BF} = \int d^{5}x {\cal L}_{\rm BF} \sim S_{\rm EH} = {1 \over 2} \int \mathrm{d}^5x R \sqrt{-g}.</math>
where <math>g=\det(g_{\mu\nu})</math> is the determinant of the [[metric tensor]] matrix, and <math>R</math> is the [[Ricci scalar]].

== Dual gravitoelectromagnetism ==

In similar manner while we define [[gravitoelectromagnetism]] for the graviton, we can define electric and magnetic fields for the dual graviton.<ref name="Henneaux">
{{cite journal
 |last1=Henneaux |first1=M.
 |last2=Teitelboim |first2=C.
 |date=2005
 |title=Duality in linearized gravity
 |journal=Physical Review D
 |volume=71 |issue=2 |page=024018
 |bibcode=2005PhRvD..71b4018H
 |doi=10.1103/PhysRevD.71.024018
 |arxiv=gr-qc/0408101|s2cid=119022015
}}</ref> There are the following relation between the gravitoelectric field <math>E_{ab}[h_{ab}]</math> and gravitomagnetic field <math>B_{ab}[h_{ab}]</math>  of the graviton <math>h_{ab}</math> and the gravitoelectric field <math>E_{ab}[T_{abc}]</math> and gravitomagnetic field <math>B_{ab}[T_{abc}]</math> of the dual graviton <math>T_{abc}</math>:<ref name="Henneaux2">Henneaux, M., "''E''<sub>10</sub> and gravitational duality"
https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_workshops_conferences/jointerc_2014/henneaux.pdf</ref><ref name="Danehkar" />
: <math>B_{ab}[T_{abc}]=E_{ab}[h_{ab}]</math>
: <math>E_{ab}[T_{abc}]=-B_{ab}[h_{ab}]</math>
and [[scalar curvature]] <math>R</math> with dual scalar curvature <math>E</math>:<ref name="Henneaux2"/>
: <math>E=\star R</math>
: <math>R=-\star E</math>
where <math>\star</math> denotes the [[Hodge dual]].

== Dual graviton in conformal gravity ==

The free (4,0) [[conformal gravity]] in ''D'' = 6 is defined as
: <math>\mathcal{S}=\int \mathrm{d}^6x \sqrt{-g} C_{ABCD}C^{ABCD},</math>
where <math>C_{ABCD}</math> is the [[Weyl tensor]] in ''D'' = 6. The free (4,0) conformal gravity can be reduced to the graviton in the ordinary space, and the dual graviton in the dual space in ''D'' = 4.<ref name="Hull2">
{{cite journal
 |last1=Hull |first1=C. M.
 |date=2000
 |title=Symmetries and Compactifications of (4,0) Conformal Gravity
 |journal=[[Journal of High Energy Physics]]
 |volume=2000 |issue=12 |page=007
 |arxiv=hep-th/0011215
 |bibcode=2000JHEP...12..007H
 |doi=10.1088/1126-6708/2000/12/007
 |s2cid=18326976
}}</ref>

It is easy to notice the similarity between the [[Lanczos tensor]], that generates the Weyl tensor in geometric theories of gravity, and Curtright tensor, particularly their shared symmetry properties of the linearized spin connection in Einstein's theory. However, Lanczos tensor is a tensor of geometry in D=4,<ref>{{cite journal |last1=Bampi |first1=Franco |last2=Caviglia |first2=Giacomo |date=April 1983 |title=Third-order tensor potentials for the Riemann and Weyl tensors |journal=General Relativity and Gravitation |language=en |volume=15 |issue=4 |pages=375–386 |doi=10.1007/BF00759166 |issn=0001-7701 |bibcode=1983GReGr..15..375B |s2cid=122782358 }}</ref> meanwhile Curtright tensor is a field tensor in arbitrary dimensions.

== See also ==
* [[Graviton]]
* [[Gravitino]]
* [[Gravity]]
* [[Gravitoelectromagnetism]]
* [[Curtright field]]
* [[Taub–NUT space]]
* [[Nielsen–Olesen vortex]]
* [['t Hooft loop]]
* [[Dual photon]]
* [[Massive gravity]]
* [[Horndeski's theory]]
* [[List of hypothetical particles]]

== References ==
{{reflist|30em}}

{{Particles}}
{{quantum gravity}}
{{String theory topics |state=collapsed}}
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